How do you find the integral of #dx/ sqrt(x^2 - a^2)#?

Answer 1
With this general case, notice how #sqrt(x^2 - a^2) prop sqrt(sec^2theta - 1)#. So, we can use the following substitution:
#x = asectheta# #dx = asecthetatanthetad theta# #sqrt(x^2 - a^2) = sqrt(a^2sec^2theta - a^2) = atantheta#

Thus, we have:

#= int 1/(cancel(atantheta))*cancel(a)secthetacancel(tantheta)d theta#
#= int secthetad theta#
Then just a little trick: #= int sectheta((sectheta + tantheta)/(sectheta + tantheta))d theta#
#= int (sec^2theta + secthetatantheta)/(sectheta + tantheta)d theta#
Now, let: #u = sectheta + tantheta# #du = secthetatantheta + sec^2thetad theta#

Therefore:

#= int1/udu#
#= ln|u|#
#= ln|sectheta + tantheta|#
Recall that: #sectheta = x/a# #tantheta = sqrt(x^2 - a^2)/a#

Thus we have:

#= color(green)(ln|x/a + sqrt(x^2 - a^2)/a| + C)#

...which is perfectly acceptable. But, you could simplify this more and be a little sneaky.

#= ln|(1/a)[x + sqrt(x^2 - a^2)]| + C#
#= ln|x + sqrt(x^2 - a^2)| + ln|1/a| + C#
but since #a# is a constant... it gets embedded into #C#.
#= color(blue)(ln|x + sqrt(x^2 - a^2)| + C)#

So if you see Wolfram Alpha give you this answer, that's why.

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Answer 2

To find the integral of ( \frac{dx}{\sqrt{x^2 - a^2}} ), where ( a ) is a constant, you can use a trigonometric substitution. Let ( x = a \sec(\theta) ), then ( dx = a \sec(\theta) \tan(\theta) d\theta ). Substituting these into the integral, you get ( \int \frac{a \sec(\theta) \tan(\theta) d\theta}{\sqrt{a^2 \sec^2(\theta) - a^2}} ). Simplifying the expression under the square root gives ( \int \frac{a \sec(\theta) \tan(\theta) d\theta}{\sqrt{a^2(\sec^2(\theta) - 1)}} ), which becomes ( \int \frac{a \sec(\theta) \tan(\theta) d\theta}{\sqrt{a^2 \tan^2(\theta)}} ). This simplifies further to ( \int d\theta ). After integrating with respect to ( \theta ), you can convert back to the original variable ( x ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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